A method of combined least squares and non-linear optimisation for the interpretation of magnetic anomalies over dykes is here presented. The method seeks to minimise a non-linear objective function by iteratively varying the non-linear parameters of the dyke while obtaining optimum values of the linear parameters by least squares analysis until an acceptable fit is obtained between the observed and computed anomalies. A study of the effects of demagnetization in arbitrarily shaped bodies and methods for evaluating the demagnetization effects of such bodies are also presented. Sixteen profiles have been taken across a linear magnetic feature which intersects the North Minch on the North Scottish Shelf. These have been interpreted in terms of a dyke about 1 km wide using the non-linear optimization techniques developed. The dyke is reversely magnetized in a direction consistent with its Tertiary origin. An aeromagnetic study of the Lower and Middle Benue Trough of Nigeria has been carried out. Regions of high and low magnetic anomalies have been correlated in an effort to find trends. Two-dimensional interpretation of several aeromagnetic profiles across the trough has been carried out. Interpretation of the observed anomalies in terms of topographic variations of the basement led to rather unreasonable models. The anomalies were best interpreted in terms of basic intrusive bodies which could occur either predominantly within the Cretaceous sediments or within the metamorphic basement. The model intrusives have variable thicknesses and directions of magnetization, suggesting that although derived from the same basic mantle material, the intrusions were emplaced at different polarity epochs. An attempt is also made to explain the tectonic evolution of the trough in terms of the models obtained.